Cauchy Residue Theorem and K-residue Theorem

Abstract

Due to negative numbers do not have square roots, scientists introduced complex numbers, which are more abstract compared with real numbers. Residue Theorem has a very significant status in complex analysis – it can be used to simplify difficult integrals. In this article, Cauchy’s Residue Theorem is first introduced with definition and proof. Then the author shows two examples of using the theorem to solve problems, which can help readers to better explain the theorem. After that, K-residue Theorem is introduced, which is a special kind of Residue Theorem, with a special function called K-function. Finally, the theorem is used to solve a problem. These problems are difficult to solve without Residue Theorem. Through the analysis, it is easy to find the importance of Residue Theorem: it is very useful, not only in math but also in many other fields like physics, biology, and chemistry. Moreover, it still has more potential for people to develop.